The colon-equals should be used, if at all, only for definitions. I don’t use it; I think that it’s always pretty clear from context when an equals sign is a definition and when it’s a statement. (On the very rare occasions when I use a special symbol, I use $\triangleq$.)
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Brian M. ScottAug 13 '12 at 15:20

As the book of Walter Rudin teaches, the best idea is always to write in the correct language what you are doing. Too many symbols mean a waste of time. Use words whenever possible.
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SiminoreAug 13 '12 at 15:41

2

@BrianM.Scott: The problem is, sometimes you're not sure if the author is making a local definition, or just using a symbol you're not familiar with (or a previous local definition you have since forgotten) and affirming an equality, which can be quite confusing. I sometimes wish people used $:=$ more often. In any event, the symbol is also used for variable assignment in Pascal and pseudocode (and maybe others, I'm not much of a programmer).
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tomaszAug 13 '12 at 16:02

6 Answers
6

An "equality by definition" is a directed mental operation, so it is nonsymmetric to begin with. It's only natural to express such an equality by a nonsymmetric symbol such as $:=\, .\ $ Seeing a formula like $e=\lim_{n\to\infty}\left(1+{1\over n}\right)^n$ for the first time a naive reader would look for an $e$ on the foregoing pages in the hope that it would then become immediately clear why such a formula should be true.

On the other hand, symbols like $=$, $\equiv$, $\sim$ and the like stand for symmetric relations between predeclared mathematical objects or variables. The symbol $\equiv$ is used , e.g., in elementary number theory for a "weakened" equality (equality modulo some given $m$), and in analysis for a "universally valid" equality: An "identity" like $\cos^2 x+\sin^2 x\equiv1$ is not meant to define a solution set (like $x^2-5x+6=0$); instead, it is expressing the idea of "equal for all $x$ under discussion".

It's entirely up to the whim of the author. Other symbols that can mean the same thing are $\triangleq$ and $=_{def}$. I think that only a minority of authors use any special notation, however; the majority just use a regular equals sign.

Sometimes, I even see $\stackrel{\text{def}}{=}$...
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Guess who it is.Aug 13 '12 at 15:54

I prefer that notation, as the delta equals symbol seems to not be universal, and the colon equals symbol can get munged with assignment in Maple/Pascal, but this symbol is unambiguous.
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ArkamisAug 13 '12 at 16:03

1

But many languages use $=$ for assignment. Should I therefore avoid $=$ in equations because it could be mistaken for a C/Java/Perl/Python/... assignment?
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celtschkAug 13 '12 at 17:29

The notation $x:= y$ is preferred as $\equiv$ has another meaning in modular arithmetic (though it is almost always clear from context as to which is meant). However, there is one big advantage to using the $:=$. That is, it is not graphically symmetric and hence allows for strings such as
$$
y:= f(x) \leq g(x) =: L
$$
where here we are defining both $y$ and $L$. This statement would be much more cumbersome using $\equiv$, and it would not make sense if one simply wrote
$$
y \equiv f(x) \leq g(x) \equiv L.
$$

Upvoting the other answers and comments... and: conventions vary. The only way to know with reasonable confidence is from context. However, one can't know whether an author "believes in" setting context. The most important criterion may be whether or not one is tracking things well enough to reasonably infer which equalities are assignments, and which are assertions. If this seems to be an issue, likely one should back-track a bit, anyway.

In practice, assignment will be clear because the left-hand side is a single symbol (even if composed of several marks), and is appearing for the first time. The first-time-appearance criterion is obviously more easily applied if all first-time appearances are highlighted by a consistent convention (rather than being buring in-line, without emphasis or fanfare).

The $\equiv$ symbol has different standard meanings in different contexts:

Congruence in number theory, and various generalizations;

Geometric congruence;

Equality for all values of the variables, as opposed to an equation in which one seeks the values that make the equation true;

$x$ is defined to be $y$;

probably a bunch of others.

But I suspect "$:=$" is not used for anything other than definitions.

So the latter at least avoids ambiguity. But if you're reading something written by someone who doesn't see it that way, you still want to understand what is being said, so you should be aware of usage conventions that you might reasonably consider less than optimal.